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Unformatted text preview: EE263 S. Lall 2011.04.07.02 Homework 2 Solutions Due Thursday 4/14 at 5 PM on the filing cabinets at the second floor of Packard. 1. Solving triangular linear equations. 42085 Consider the linear equations y = Rx , where R ∈ R n × n is upper triangular and invertible. Suggest a simple algorithm to solve for x given R and y . Hint: first find x n ; then find x n- 1 (remembering that now you know x n ); then find x n- 2 (remembering that now you know x n and x n- 1 ); etc. Remark: the algorithm you will discover is called back substitution . It requires order n 2 floating point operations (flops); most methods for solving y = Ax for general A ∈ R n × n require order n 3 flops. Solution. Suppose that y = y 1 y 2 . . . y n , R = r 11 r 12 ··· r 1 n r 22 ··· r 2 n . . . . . . . . . ··· r nn , x = x 1 x 2 . . . x n . Consider the linear equation corresponding to the last ( n th) row of R , i.e. , y n = r nn x n . If r nn negationslash = 0 we can simply solve for x n to get x n = y n /r nn . If r nn = 0 then two cases may occur. Either y n negationslash = 0 which implies that the set of linear equations is inconsistent or y n = 0 which implies that the choice of x n is arbitrary. In any case, r nn = 0 means that a unique solution does not exist for the set of linear equations. Now consider the linear equation corresponding to the ( n − 1)th row of R , i.e. , y n- 1 = r ( n- 1)( n- 1) x n- 1 + r ( n- 1) n x n and for r ( n- 1)( n- 1) negationslash = 0 we get x n- 1 = 1 r ( n- 1)( n- 1) ( y n- 1 − r ( n- 1) n x n ) with x n found from the previous step. Again if r ( n- 1)( n- 1) = 0 it can be said that the system of linear equations has no unique solution. In general, if x n , x n- 1 , . . . , x i +1 are known, x i can be derived from the linear equation corresponding to the i th row of R as (assuming r ii negationslash = 0) x i = 1 r ii ( y i − r i ( i- 1) x i- 1 − r i ( i- 2) i x i- 2 − ··· − r in x n ) . Therefore, the x i ’s can be computed recursively for i = n, n − 1 , . . . , 1 by back substitution . This suggests the following simple algorithm: i := n ; while i ≥ 1 if r ii negationslash = 0 x i := 1 r ii parenleftBig y i − ∑ n j = i +1 r ij x j parenrightBig ; else unique solution does not exist; break ; end i := i − 1; end 1 EE263 S. Lall 2011.04.07.02 Note that whenever r ii = 0 a solution does not exist or the solution is not unique. We know that the condition for a (unique) solution to exist is det R = producttext n i =1 r ii negationslash = 0 which also implies that none of the diagonal elements r ii of R to be zero....
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This note was uploaded on 07/28/2011 for the course EE 263 taught by Professor Boyd,s during the Summer '08 term at Stanford.
- Summer '08